SCIENCE 



NEW YORK, MAY 20, 1892. 



THE GROWTH OF CHILDREN.— II. 

 In No. 483 of Science I have tried to show that measurements 

 of children of a given age are, as a rule, not distributed sym- 

 metrically around the average, but that they are distributed 

 asymmetrically, the curve being expressed by the formula — 



(1 4- C 11)^ 



-3 M = 



In this expression c is a small constant, M the mean variation, and 

 u the deviation from that measurement which belongs to the in- 

 dividual which will finally be an average individual in regard to 

 the measurement under consideration and whose development 

 corresponds exactly to that of its age. In this sense the measure- 

 ment may be called that of the average individual, although it is 

 not the average of all the measurements. 



Supposing an extensive series of observations on children of a 

 certain age to be given, the question arises, how to find that value 

 which belongs to the average individual and how to find the mean 

 variation. The number of observations between the limits a and 

 h will be 



h 



f. 





iW" 1/2 



cM (e 



V 2n 



2M--B 'iM' 



Whenever o and b remain the same multiples of M, the value of 

 cM 



this integral depends solely on 



V2: 



and a table of the values of 



the integral may be computed. It is convenient to assume a = 

 — 00 and to compute the integral. Following is a brief table of 

 the integral: — 



M V 2 



fh' 



V 2 



c u 



(/ 2 7i 



-3.0Jlf 

 -2.5itf 



-0.10 -0.08 -0.06 -0.04 -0. 



0.0025 

 0,0106 



0.0023 

 0.0097 



0.09250.( 

 209110.) 



1.0018 0.0016 

 1.0080 0.0071 



0.00 +0.02 +".04 -f0.06 -fO.08 +0.10 



0.0012 

 0.0053 



0.0010 

 0.0044 



0.0007 

 0.0036 



0.0005 

 0.0027 



0.0003 

 0.0018 



0.0254 

 0.0731 

 0,1727 



1.0666 0. 



0173 0.0146 

 0637 0.0472 

 1363 0.1242 



0.01190.0091 

 0.0407:0.0342 

 0.11210.1000 



-O.Silf 



0.0 Jlf 



+ 0.5 4f 



0.6000!0.5800 

 0.779710.7631 



+ 1.0itf 

 + 1.5 M 

 + 2.0 M 



0.9OWJ0.8879 

 0.96590.9593 

 0.99080.9881 



+ 2.6 Jf 

 + 3.0 3f 



0.993810.9973 

 0.999710.9995 



1.343810.32610, 

 i.5400|0.5200 0, 



3085 0. 

 5000:0. 

 69150. 



,2909 0. 

 ,4800 0, 

 6739 0. 



,2732 0.2556 0.2379 0.2303 



I I 

 4600 0.4400 0.4200 0.4000 



.6562 0.6386'o.6209 0.6033 



I i I 



1.8758 0.8637|0.8515;0.8394 

 1.9528 

 I.9bff4 0. 



) 9463 0.9399 0.£ 



0.9947 

 0.9988 



),8151 0.8030 0,7909 0.7788 

 ).9205'0.9140 0.9075 0.9010 

 ).9719'o,9693 1 .9665 0.9637 



1.992010.9912 



1.9982 0.9979 



0.9903 

 0.9977 



The series of actual observations/mustl^correspondjto one of 

 these theoretical curves. We must find those values of c and M 

 which agree most nearly with the curve of the observations, c 

 and M may be determined from any two values of the integral. 

 The most probable values will be those which are found by taking 

 into consideration all the given values. This may be done in the 

 following way : We will call the value for which u = 0, U; then, 

 any observed value 



Y=U+u. 

 The average of all observed values 



^.=/i^ 



+ M) (1 + eu) 



31 ViTT 



2M-'au 



(1) Ai= U + cM^ 



The average of the squares of all observed values 



+ 3J 



'+U)' (1 +CM)^ 



1/ t^ arr 



= fJ" -I- 2 ZJcJM^ + Jlf = 



= Jja -I- 2 U"(Ai - C/) + ilf 2 



= - U^ +%U A^ + M^\ smA 



(4) 



) U =A^± s/M^ -{A^-A-,^). 



By substituting this value in (1) we find 

 cM 



T \/ M^ -{A^- A^^) 



^' 277 " ilf ^' 27r 



By computing the average of the observations and of their 

 squares, we can, therefore, find easily a series of the three values 

 fJi, M, c, and we have to select the one which gives the most sat- 

 isfactory agreement between the theoretical curve and the actual 

 curve, i.e., the one in which the sum of the squares of the differ- 

 ences between the two curves are a minimum. The actual com.- 

 putation becomes a little simpler by substituting 



Y= C -\-y where C is equal or nearly equal Ai. 

 The average of all 2/ a^ = JJ- C + c M^ =0 

 The average of all 2/= a^ = (U- C)^ + 2{U - C){Ai- U)+M' 

 = {C-Ai)^-{U- A,) '- + M^ 

 U = A^± V iM^^^oT+W^^^O^ 



I wall show the application of this method by computing the 

 stature of 12-year-old girls, measured in Worcester, Mass,, 1891. 

 112 observations are available. 



^1 = 1446.6; 0=1447; a^ = 5365. 

 U = 1446.6 ± vji/ 2 -5364.84. 



We assume various values for M, and find the corresponding 

 cJlf 



values for U and 



^' 37r 



Then the number of cases which are required by the theory 

 may be found from the above table, while the observed number 

 of cases are found by computing U — Z M, U — 2.^ M, etc., and 



